<i>The Stability of Motion</i>

“…To show that P*(i, v) -> » as | i \ + | v | -> , we write P* in the form P*(i, v) = i(P, , B~lP,) + W(i), where 13) and proceed as in the proof of theorem 3. We remark that it is not obvious, in general, when the first two conditions of theorems 3 and 4 hold.…”

Section: = C(v)mentioning

confidence: 99%

A theory of nonlinear networks. I

Brayton¹,

Moser²

1964

Quart. Appl. Math.

396

240

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“…The fundamental classical stability theorems ͑Thom-son and Tait, 1879; Chetayev, 1961;Merkin, 1997͒, some of which will be discussed in this review, can be deduced either by appealing to spectral properties of the dynamical system ͑2.4͒ or by appealing to the geometrical properties of Eq. ͑2.6͒.…”

Section: ͑29͒mentioning

confidence: 99%

Dissipation-induced instabilities in finite dimensions

2007

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The goal of this work is to introduce a coherent theory of the counterintuitive phenomena of dynamical destabilization under the action of dissipation. While the existence of one class of dissipation-induced instabilities was known to Sir Thomson ͑Lord Kelvin͒, it was not realized until recently that there is another major type of these phenomena hinted at by one of Merkin's theorems; in fact, these two cases exhaust all the generic possibilities. The theory grounded on the Thomson-Tait-Chetayev and Merkin theorems and on the geometric understanding introduced in this paper leads to the conclusion that ubiquitous dissipation is one of the paramount mechanisms by which instabilities develop in nature. Along with a historical review, the main theoretical achievements are put in a general context, thus unifying the current knowledge in this area and the multitude of relevant physical problems scattered over a vast literature. This general view also highlights the striking connection to various areas of mathematics. To appeal to the reader's intuition and experience, a large number of motivating examples are provided. The paper contains some new unpublished results and insights, and, finally, open questions are formulated to provide an impetus for future studies. While this review focuses on the finite-dimensional case, where the theory is relatively complete, a brief discussion of the current state of knowledge in the infinite-dimensional case, typified by partial differential equations, is also given.

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“…The phenomenon of dissipation-induced instabilities in finite-degree of freedom mechanical systems was known at the end of the nineteenth century to Thomson and Tait [83], who made it precise by formulating theorems which were later proved by Chetayev [16]. The corollary of these theorems relevant to our discussion states that if a system with an unstable potential energy has been (spectrally) stabilized with gyroscopic forces, then this stability is lost after the addition of arbitrarily small dissipation.…”

Section: Introductionmentioning

confidence: 91%

Dissipation-Induced Instability Phenomena in Infinite-Dimensional Systems

Krechetnikov

Marsden

2009

Arch Rational Mech Anal

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This paper develops a rigorous notion of dissipation-induced instability in infinite dimensions as an extension of the classical concept implicitly introduced by Thomson and Tait for finite degree of freedom mechanical systems over a century ago. Here we restrict ourselves to a particular form of infinite-dimensional systems-partial differential equations-whose inherent function-analytic differences from finite-dimensional systems make uncovering this notion more intricate.

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The Stability of Motion

Cited by 51 publications

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A theory of nonlinear networks. I

A theory of nonlinear networks. I

Dissipation-induced instabilities in finite dimensions

Dissipation-Induced Instability Phenomena in Infinite-Dimensional Systems

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